Just a quick look: $\vec{T}_s$ is wrong. It should be $\vec{T}_s = (1,0,-s(a^2-s^2-t^2)^{-1/2})$ (and similarly for $\vec{T}_t$).

I took a closer look. It won't affect your calculations. You have:

$\displaystyle \int\int_D\left[(s^2+t^2)(a^2-s^2-t^2)^{-1/2} + (a^2-s^2-t^2)^{1/2}\right]dsdt$

I will rewrite this:

$\dfrac{s^2+t^2}{(a^2-s^2-t^2)^{1/2}} + (a^2-s^2-t^2)^{1/2} = \dfrac{s^2+t^2 + (a^2-s^2-t^2)}{(a^2-s^2-t^2)^{1/2}}$

Simplifying, you get:

$\dfrac{a^2}{(a^2-s^2-t^2)^{1/2}}$

So, your whole integral simplifies to:

$\displaystyle \int\int_D \dfrac{a^2}{(a^2-s^2-t^2)^{1/2}}dsdt$

Does that help?